Assigning vertex groups to bones precisely
Fast and easy way to do it is in edit mode, select the part of the mesh that you want to be effected buy one bone, then in vertex groups select the group of related bone and assign weight with value of 1.

Then select other other vertex group and hit remove button.

Then repeat the same process on the other part of the mesh assigning or removing weight that you want to be associated with that part of the mesh.